HCU MCA 2013 Paper

Question : 52 of 75

Marks: +1, -0

If a, b and c are all > 0, and are in a geometric progression as the successive terms ...,p, q, r, ... Then, what is the value of the determinant D below?

D = |\table \log \;a,p,1;\log\;b,q,1;\log\;c,r,1|

0
pqr
1
pqr - abc

Validate

Solution:

(a) Let first term of GP is A and common ratio is R.
Then, a = AR

^{n-1}

⇒ log a = log A + (P - 1) log R etc.
⇒

|\table \log \;a,p,1;\log\;b,q,1;\log\;c,r,1|=|\table \log\;\A+(p-1)\log\;\R,p,1;\log\;\A+(q-1)\log\;\R,q,1;\log\;\A+(r-1)\log\;\R,r,1|

=|\table (p-1)\log\;\R,p,1;(q-1)\log\;\R,q,1;(r-1)\log\;\R,r,1|\;\;\;\;\;\;\;\;[∴ \C_1→\C_1-(\lo g \A)C_3]

=\log \;\R|\table (p-1),p,1;(q-1),q,1;(r-1),r,1|

=\log \;\R|\table p,p,1;q,q,1;r,r,1|\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;[∴ \C_1→\C_1+\C_3]

= 0

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